### 统计代写|蒙特卡洛方法代写Monte Carlo method代考|MAT 359

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Definition of Models for Emission, Absorption

A surface is formally defined in Section $1.4$ as the interface between two regions of space having different optical properties, where the optical properties in question are the refractive and absorptive indices $n$ and $k$. We distinguish between these two properties and the models used to characterize the interaction between thermal radiation and surfaces. We are now in a position to elaborate on the definition and use of the surface

interaction models for emission, reflection, and absorption introduced in Chapter $1 .$

We define the directional spectral emissivity $\varepsilon(\lambda, T, \vartheta, \varphi)$ as the ratio of the spectral intensity of emission from a real body in direction $(\vartheta, \varphi)$ to the spectral intensity of a blackbody at the same temperature,
$$\varepsilon(\lambda, T, \vartheta, \varphi) \equiv \frac{i_{\lambda, e}(\lambda, T, \vartheta, \varphi)}{i_{b \lambda}(\lambda, T)}$$
Note that the symbol for the directional spectral emissivity can also be written $\varepsilon_{\lambda}^{\prime}(T)$, where the prime $\left({ }^{\prime}\right)$ indicates directionality and the subscript $\lambda$ identifies the model as spectral.

The directional total emissivity of a surface is then related to the directional spectral emissivity according to
$$\varepsilon^{\prime}(T)=\varepsilon(T, \vartheta, \varphi)=\frac{\int_{\lambda=0}^{\omega} \varepsilon(\lambda, T, \vartheta, \varphi) i_{b \lambda}(\lambda, T) d \lambda}{\int_{\lambda=0}^{\infty} i_{b \lambda}(\lambda, T) d \lambda} .$$
We know that the denominator is $\sigma T^{4} / \pi$, so Eq. (2.31) can be rewritten
$$\varepsilon^{\prime}(T)=\varepsilon(T, \vartheta, \varphi)=\frac{\pi}{\sigma T^{4}} \int_{\lambda=0}^{\infty} \varepsilon(\lambda, T, \vartheta, \varphi) i_{b \lambda}(\lambda, T) d \lambda .$$
A surface is said to be gray in a given direction $\left(\vartheta_{1}, \varphi_{1}\right)$ if the directional spectral emissivity is independent of wavelength in that direction. Equation (2.32) then defines a gray equivalent directional emissivity for spectral surfaces. The spectral intensities of a blackbody, a graybody, and a hypothetical real surface, all at $6000 \mathrm{~K}$, are compared in Figure $2.11$.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Introduction to the Radiation Behavior of Surfaces

The primary surface conditions that influence the radiation behavior of a solid surfacé ăré its bulk êlectricall propertiês (èléctrical conductor or non-conductor), its topography (smooth, polished, sanded, sand blasted, turned, lapped, honed, ground, peened, etc.), its chemical condition (reduced, oxidized, anodized, galvanized, etc.), its degree of contamination (clean or dirty, dusty, dry or oily, etc.), and its surface grain structure (annealed, cold rolled, hot rolled, etc.). Surfaces may also be painted, sputter coated, or evaporation coated to enhance or diminish emission, absorption, or reflection or to bias directionality and/or wavelength dependence. In addition, all surface preparations are subject to damage and aging. The result is a bewilderingly subjective array of adjectives, often open to interpretation, which renders effective communication between designers and modelers difficult if not impossible. Still, it is imperative, once a project moves out of the preliminary design phase, that engineers charged with performance modeling have access to accurate models for surface radiation behavior. In the extreme this often requires a surface characterization campaign, typically based on measurement of the bidirectional spectral reflectivity of key surfaces. The following brief review is intended as a guide to the reader tasked with formulating surface radiation behavior models, a topic treated in more detail in Chapter $4 .$
Maxwell’s Equations [10],
$$\nabla \times \boldsymbol{H}=\varepsilon_{0} \frac{\partial \boldsymbol{E}}{\partial t}+\frac{\boldsymbol{E}}{r_{e}}$$

$$\begin{gathered} \nabla \times \boldsymbol{E}=-\mu_{0} \frac{\partial \boldsymbol{H}}{\partial t} \ \nabla \cdot \boldsymbol{E}=\frac{\rho_{e}}{\varepsilon} \ \nabla \cdot \boldsymbol{H}=0 \end{gathered}$$
are the point of departure for understanding the interaction of EM radiation with a surface. In Eqs. (2.62) to (2.65), $\boldsymbol{H}\left(\mathrm{Am}^{-1}\right)$ is the magnetic field strength, $\mathrm{E}\left(\mathrm{V} \mathrm{m}^{-1}\right)$ is the electric field strength, $\varepsilon_{0}=8.854 \mathrm{C}^{2} / \mathrm{N} \cdot \mathrm{m}^{2}$ is the permittivity of free space, $r_{e}(\Omega-\mathrm{m})$ is the electrical resistivity, $\rho_{e}\left(\mathrm{C} \mathrm{m}^{-3}\right)$ is the electric charge density, and $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~N} \mathrm{~A}^{-2}$ is the magnetic permeability. These celebrated equations were formulated in 1864 by the British physicist and mathematician James Clerk Maxwell, who synthesized them from already known relationships between electricity and magnetism. Their solution permitted for the first time the theoretical calculation (see Problem 2.7) of the already known speed of light in a vacuum, thereby removing any doubt as to their validity.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Radiation Behavior of Surfaces Composed

In the following paragraphs we consider the two limiting cases in which a monochromatic EM wave is incident to the plane interface separating two ideal regions. In the first extreme, the wave passes from a dielectric whose optical properties are $n_{1}$ and $k_{1}=0\left(r_{e} \rightarrow \infty\right)$ into another dielectric whose optical properties are $n_{2}$ and $k_{2}=0$; in the second extreme, the wave passes from a dielectric whose optical properties are $n_{1}$ and $k_{1}=0$ into an electrical conductor, or metal, whose optical properties are $n_{2}$ and $k_{2} \neq 0$

The case of a smooth plane interface between two dielectrics with $n_{2}>n_{1}$ is illustrated in Figure 2.19. In the figure $\boldsymbol{E}_{p, i}$ represents the

eléctric component of a transverse-magnetic (TM), p-polarized, monochromatic electromagnetic wave incident to the interface, or surface. The arrows labeled “Incident,” “Reflected,” and “Transmitted” can be thought of as “rays,” in which case the lines passing normal to the rays indicate wavefronts separated by a distance $\lambda$. Following convention, the subscript ” $p$ ” is used to remind us that the electric field vector in this case lies in the plane of incidence, the plane containing both the incident ray and the unit normal vector $\boldsymbol{n}=-i$. Without loss of generality we consider only the real part of the incident electric field,
$$\operatorname{Re}\left[E_{p, i}\right]=\left|E_{p, i}\right| \cos \left[\omega\left(\frac{n_{1} x^{\prime}}{c_{0}}-t\right)\right],$$
where $\omega=2 \pi c_{0} / \lambda$ and $x^{\prime}=y / \sin \vartheta_{i}$. This is equivalent to assuming that the phase angle of the incident wave, $\phi_{i}=\tan ^{-1}\left[\operatorname{Im}\left(E_{p, i}\right) / \operatorname{Re}\left(E_{p, i}\right)\right]$, is zero.

Careful consideration of Figure $2.19$ reveals that the $y$-component (parallel to the interface) of the electric field above the interface (region 1) is
\begin{aligned} \left|E_{y, 1}\right|=&\left|E_{p, i}\right| \cos \vartheta_{i} \cos \left[\omega\left(\frac{n_{1} y / \sin \vartheta_{i}}{c_{0}}-t\right)\right] \ &-\left|E_{p, r}\right| \cos \vartheta_{r} \cos \left[\omega\left(\frac{n_{1} y / \sin \vartheta_{r}}{c_{0}}-t\right)\right] \end{aligned}

and the $y$-component below the interface (region 2) is
$$\left|E_{y, 2}\right|=\left|E_{p, t}\right| \cos \vartheta_{t} \cos \left[\omega\left(\frac{n_{2} y / \sin \vartheta_{t}}{c_{0}}-t\right)\right] .$$

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Definition of Models for Emission, Absorption

e(λ,吨,ϑ,披)≡一世λ,和(λ,吨,ϑ,披)一世bλ(λ,吨)

e′(吨)=e(吨,ϑ,披)=∫λ=0ωe(λ,吨,ϑ,披)一世bλ(λ,吨)dλ∫λ=0∞一世bλ(λ,吨)dλ.

e′(吨)=e(吨,ϑ,披)=圆周率σ吨4∫λ=0∞e(λ,吨,ϑ,披)一世bλ(λ,吨)dλ.

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Introduction to the Radiation Behavior of Surfaces

∇×H=e0∂和∂吨+和r和

∇×和=−μ0∂H∂吨 ∇⋅和=ρ和e ∇⋅H=0

## 统计代写|蒙特卡洛方法代写Monte Carlo method代考|Radiation Behavior of Surfaces Composed

|和是,1|=|和p,一世|因⁡ϑ一世因⁡[ω(n1是/罪⁡ϑ一世C0−吨)] −|和p,r|因⁡ϑr因⁡[ω(n1是/罪⁡ϑrC0−吨)]

|和是,2|=|和p,吨|因⁡ϑ吨因⁡[ω(n2是/罪⁡ϑ吨C0−吨)].

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